Forum:Did Frodo and Bilbo go to Valinor or Tol Eressëa?

I'm sure that this discussion has been had before, and I suspect from the outset that we don't know the answer. It is said at the end of The Lord of the Rings,

And the ship went out into the High Sea and passed on into the West, until at last on a night of rain Frodo smelled a sweet fragrance on the air and heard the sound of singing that came over the water. And then it seemed to him that as in his dream in the house of Bombadil, the grey rain-curtain turned all to silver glass and was rolled back, and he beheld white shores and beyond them a far green country under a swift sunrise.—Narrator[1]

[...]they should always be able to leave Middle-earth, if they wished, and pass over Sea to the True West, by the Straight Road, and so come to Eressëa – but so pass out of time and history, never to return.—J.R.R. Tolkien[2]

By implication this would mean that those who travel along the Straight Road come first to Eressëa, and perhaps the 'white shores and beyond them a far green country' refers to Eressëa.

In Letter 325 Tolkien writes about Frodo and travelling into the West,

As for Frodo or other mortals, they could only dwell in Aman for a limited time – whether brief or long.—Tolkien[3]

Earlier in the same letter Tolkien wrote,

The 'immortals' who were permitted to leave Middle-earth and seek Aman – the undying lands of Valinor and Eressëa, an island assigned to the Eldar—Tolkien[4]